Quasilinearization Technique for -Laplacian Type Equations

Abstract

An equation is considered together with the boundary conditions , . This problem under appropriate conditions can be reduced to quasilinear problem for two-dimensional differential system. The conditions for existence of multiple solutions to the original problem are obtained by multiply applying the quasilinearization technique.

New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

“…Thus, assumption (21) proves desired inequality (19), and thus, Corollary 2 verifies that (20) is oscillatory provided (21) holds.…”

“…Firstly, it is elementary to check that the required inequalities (12) and (13) are satisfied. Next, we prove that required inequality (19) immediately follows from assumption (21). On the first hand, we estimate from above the term on the right hand side in (19) using that 1 1 + 2 2 = 1:…”

“…On an interesting example in [5], the author established the oscillations provided at least one of constants appearing in the coefficients is sufficiently large (see also [6] for = 1), which is presented here in the next section as a particular case of our main results. On the properties of some classes of the one-dimensional quasilinear differential equations with -Laplacian operators we refer reader to [14][15][16][17][18][19][20] and the references therein. About the applications of second-order functional differential equations in the mathematical description of certain phenomena in physics, technics, and biology (oscillation in a vacuum tube; interaction of an oscillator with an energy source; coupled oscillators in electronics, chemistry, and ecology; relativistic motion of a mass in a central field; ship course stabilization; moving of the tip of a growing plant; etc.…”

On Types of Solutions of the Second Order Nonlinear Boundary Value Problems

“…The applicability of the quasilinearization process was demonstrated in the papers [20][21][22][23][24].…”